Methods to distinguish continuous probability distributions

Question

I read in the Wikipedia article for Variance

The variance is one of several descriptors of a probability distribution. In particular, the variance is one of the moments of a distribution. In that context, it forms part of a systematic approach to distinguishing between probability distributions. While other such approaches have been developed, those based on moments are advantageous in terms of mathematical and computational simplicity.

I am very interessted in methods to distinguish continuous probability distributions. Especially to distinguish analytically if the distribution has two distinct peaks or just one.

Does someone know what is meant with "While other such approaches have been developed"?

Answer 1

See

http://en.wikipedia.org/wiki/Moment_(mathematics)#Variance http://en.wikipedia.org/wiki/Moment-generating_function

here the moment generating function completely determined the random variable. But the moments are not, as there are different random variables with identical moments. So the first and second moment (or the variance and the mean) is still a very weak invariant.

Answer 2

You are probably looking for a bimodality test. Read http://en.wikipedia.org/wiki/Bimodal_distribution.

Answer 3

If a rough detection suffices, I suggest this elementary scheme:

• find the interesting range of the variable (delimit the tails),
• evaluate the distribution at equally spaced points, using the smallest distinguishable separation (say 100 points),
• report any maximum configuration $f_{n-1}\lt f_n\gt f_{n+1}$.

More sophisticated schemes can be used to find changes of sign of the derivative, like finite differencing and root finding, but I am not sure they will bring any benefit.